# American Institute of Mathematical Sciences

June  2012, 32(6): 2285-2299. doi: 10.3934/dcds.2012.32.2285

## Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients

 1 Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy, Italy

Received  January 2011 Revised  July 2011 Published  February 2012

We prove heat kernel bounds for the operator $(1+|x|^\alpha)\Delta$ in $\mathbb{R}^N$, through Nash inequalities and weighted Hardy inequalities.
Citation: Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285
##### References:
 [1] A. Bazan and W. Neves, The Caffarelli-Kohn-Niremberg's inequality for arbitrary norms, preprint. [2] A. Bazan and W. Neves, The Hardy and Caffarelli-Kohn-Niremberg inequalities revised, preprint, arXiv:1007.2005v1. [3] D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, preprint, arXiv:1004.3456. [4] E. B. Davies, "One-Parameter Semigroups," London Mathematical Society Monographs, 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. [5] E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989. [6] K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolutions Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. [7] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete and Continuous Dynamical Systems, 18 (2007), 747-772. [8] G. Metafune, E. M. Ouhabaz and D. Pallara, Long time behavior of heat kernels of operators with unbounded drift terms, J. Math. Anal. Appl., 377 (2011), 170-179. doi: 10.1016/j.jmaa.2010.10.023. [9] G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear. [10] G. Metafune and C. Spina, Kernel estimates for a class of Schrödinger semigroups, Journal of Evolution Equations, 7 (2007), 719-742. doi: 10.1007/s00028-007-0338-3. [11] G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129. [12] B. Muckenhoupt, Hardy's inequalities with weights, Studia Math., 44 (1972), 31-38. [13] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6. [14] E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005. [15] F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.

show all references

##### References:
 [1] A. Bazan and W. Neves, The Caffarelli-Kohn-Niremberg's inequality for arbitrary norms, preprint. [2] A. Bazan and W. Neves, The Hardy and Caffarelli-Kohn-Niremberg inequalities revised, preprint, arXiv:1007.2005v1. [3] D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, preprint, arXiv:1004.3456. [4] E. B. Davies, "One-Parameter Semigroups," London Mathematical Society Monographs, 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. [5] E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989. [6] K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolutions Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. [7] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete and Continuous Dynamical Systems, 18 (2007), 747-772. [8] G. Metafune, E. M. Ouhabaz and D. Pallara, Long time behavior of heat kernels of operators with unbounded drift terms, J. Math. Anal. Appl., 377 (2011), 170-179. doi: 10.1016/j.jmaa.2010.10.023. [9] G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear. [10] G. Metafune and C. Spina, Kernel estimates for a class of Schrödinger semigroups, Journal of Evolution Equations, 7 (2007), 719-742. doi: 10.1007/s00028-007-0338-3. [11] G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129. [12] B. Muckenhoupt, Hardy's inequalities with weights, Studia Math., 44 (1972), 31-38. [13] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6. [14] E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005. [15] F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.
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